Periodic solutions of quaternionic-valued ordinary differential equations

Digital Object Identifier (DOI) 10.1007/s10231-004-0139-z

Annali di Matematica 185, S109–S127 (2006)

Juan Campos · Jean Mawhin

Periodic solutions of quaternionic-valued ordinarydifferential equations

Dedicated to Roberto Conti, on his 80th birthday

Received: October 27, 2003Published online: April 15, 2005 – © Springer-Verlag 2005

Abstract. This paper uses topological degree methods to prove the existence of periodicsolutions of some quaternionic-valued ordinary differential equations.

Mathematics Subject Classification (2000). 34C25, 11R52

Key words. periodic solutions – topological degree methods – quaternions

1. Introduction

Periodic solutions of nonlinear complex-valued periodic ordinary differential equa-tions with periodic coefficients have been considered by a number of authors. Inparticular, Pliss [31] and Lloyd [19,20] have discussed the structure of the set ofthe T-periodic solutions of equations of the form

z′ =n∑

j=0

pj(t)zj .

Using the classical connection with a second order linear equation, Lloyd [19] hasshown that a complex Riccati equation of the form

z′ = z2 + p(t)z + f(t) (1)

with real-valued T-periodic coefficients p and f always has two T-periodic solutionswhen f(t) < 0 for all t ∈ R. Furthermore, he has given an example of p and ffor which (1) has no T-periodic solution. Further results in this direction have beenobtained by Hassan [14,15], Gunson and Hassan [13], and Alwash and Lloyd [1].

J. Campos: Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad deGranada, 18071-Granada, Spain, e-mail: [email protected]

J. Mawhin: Institut mathematique, Universite Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium, e-mail: [email protected]

Supported by MCYT, BFM2002-01308 (Spain).

S110 J. Campos, J. Mawhin

As an application of his combination of generalized Conley’s blocks and Lef-schetz fixed point theorem [36,38–40], Srzednicki has proved in [37] that theequation

z′ = z2 + εeiωt,

(ω = 2π

T

)(2)

has at least one T-periodic solution for |ε| sufficiently small, but that the equation

z′ = z2 + εeiωt

has a T-periodic solution for all ε, where z(t) denotes the conjugate of z(t). Thislast assertion is a consequence of an existence result for the T-periodic solutionsof the more general complex-valued equation

z′ =∑

0≤k+l<r+s

bk,l(t)zkzl + ceiαt zr zs, (3)

when r + s ≥ 2 and either α = 0, r − s ≤ 1, or α = 0, r − s ≤ −1 and T isa multiple of 2π/α.

This result was extended in [25–27,21,22] using topological degree argumentscombined with Lr-estimates for the norm of the possible solutions. In particular, in[22,26,27], existence results were obtained for the T-periodic solutions of systemsof complex-valued ordinary differential equations of the form

ai(t)z′i = |zi |qzi

p + hi(t, z), (i = 1, 2, . . . , n),

with p ≥ 1, and

ai(t)z′i = |zi |qzi + hi(t, z), (i = 1, 2, . . . , n),

when the hi are of lower order at infinity than the principal terms of the respectiveequations. They imply in particular that the complex conjugate Riccati equation

z′ = z2 + f(t) (4)

has at least one T-periodic solution for any forcing term f(t). The more generalcase of Floquet boundary conditions has been considered in [28,29,39]. The caseof polynomial equations with coefficients belonging to the algebra of T-periodicfunctions with Fourier coefficients of negative index equal to zero has been recentlyinitiated by Borisovich and Marzantowicz [3,4] and developed by Taddei [41].

On the other hand, the question of the existence of a T-periodic solution to thecomplex Riccati equation

z′ = z2 + f(t) (5)

for an arbitrary T-periodic complex-valued forcing term f(t) has been raised in [25].A negative answer has been given by Campos and Ortega [6] by constructing a C∞T-periodic real function f for which (5) has no T-periodic solution. Gabdrakhmanovand Filippov [12] have found a class of forcing terms f(t) for which the same

Periodic solutions of quaternionic-valued ordinary differential equations S111

property holds. In [5], Campos has used Möbius transformations to determine allthe possible dynamics of the solutions of (5). Notice that an essential differencebetween (5) and (4) is that the autonomous part of (4) is a gradient system, whichis not the case for (5).

Miklaszewski [30] has proved that there exists some ε0 ∈ [1, 2] such that (2),with ω = 1, had no 2π-periodic solution for ε = ε0, if the sequence definedrecursively by

a1 = 1, an = 1

n

n−1∑

k=1

akan−k, (n ≥ 1), (6)

is such that

a2n < an−1an+1 for all n > 1. (7)

Miklaszewski has conjectured, on a numerical basis, that this last property is true,but he could not prove it. Very recently, Zoladek [43,44] has used the method ofholomorphic foliations to construct examples of equations of the form

z′ = zn +n−1∑

j=0

pj(eit)z j

with no 2π-periodic solutions. In particular he has shown that Miklaszewski’sconjecture is true for all sufficiently large values of n and that this asymptoticversion implies the existence of a sequence 0 < ε1 < ε2 < . . . , tending to +∞,such that (2) with ω = 1 has exactly one 2π-periodic solution if ε = ε j and no2π-periodic solution if ε = ε j .

One advantage of the complex formulation of the planar systems consideredabove is that the multiplicative structure of the complex numbers helps in obtainingthe required a priori estimates for the solutions. This has also been emphasized byRab, Kalas, Osica, and Tesarova in the study of stability and asymptotic behaviorproblems for complex Riccati equations (see, e.g., [32–34,16,17,42] and theirreferences).

The aim of this paper is to initiate a similar study of the T-periodic solutions ofquaternionic-valued first order differential equations

q′ = F(t, q) (8)

with F : [0, T ] × H→ H continous, where H denotes the set of quaternions (seeSection 2). Equation (8) corresponds, in terms of its components, to some systemsof four first order real-valued differential equations. A T-periodic solution of (8) isa solution q : [0, T ] → H of (8) such that

q(0) = q(T ). (9)

In Section 3, we consider the case of linear monomial equations

q′ = a(t)q + f(t),


where a : [0, T ] → H, f : [0, T ] → H, and we treat the corresponding semilinearcase

q′ = a(t)q + c(t, q)

in Section 4. In Sections 5 and 6, we obtain partial extensions to the quaternionicsetting of the main results of [22]. A simple consequence of our results is theexistence, for any forcing term f(t), of a T-periodic solution for the perturbedquaternionic Ginzburg-Landau-type equation

q′ = ‖q‖r q + f(t)

and for the quaternionic conjugate Riccati equation

q′ = q2 + f(t),

which corresponds to the system

q′0 = q2

0 − q21 − q2

2 − q23 + f1(t),

q′1 = −2q0q1 + f2(t),

q′2 = −2q0q2 + f3(t),

q′3 = −2q0q3 + f4(t).

In Section 7, we show, in particular for the quaternionic Riccati equation

q′ = q2 + f(t), (10)

which corresponds to the system

q′0 = q2

0 − q21 − q2

2 − q23 + f1(t),

q′1 = 2q0q1 + f2(t),

q′2 = 2q0q2 + f3(t),

q′3 = 2q0q3 + f4(t),

that, as in the complex case, there exists a real-valued forcing term f for which(10) has no T-periodic solution, but that such a solution always exists when f :[0, T ] → H has a sufficiently small norm.

2. Quaternions

We denote as usual a (real) quaternion q = (q0, q1, q2, q3) ∈ R4 by

q = q0 + q1i + q2 j + q3k,

where q0, q1, q2, q3 are real numbers and i, j, k symbols satisfying the multiplica-tion table formed by

i2 = j2 = k2 = ijk = −1, ij = − ji = k


and the other rules deduced by cyclic interchange of i, j, k. We denote the set ofquaternions by H.

We can define for the quaternion q and r = r0 + r1i + r2 j + r3k, the innerproduct, and the modulus, respectively, by

〈q, r〉 = q0r0 + q1r1 + q2r2 + q3r3,

‖q‖ = (〈q, q〉)1/2 = (q2

0 + q21 + q2

2 + q23

)1/2.

The above inner product makesH a four-dimensional real Hilbert space. Givenq = q0 + q1i + q2 j + q3k, we introduce the real part operator

q = q0

as well as the vectorial or imaginary part by

q = q1i + q2 j + q3k,

and an easy computation shows that

(q)2 = −‖q‖2. (11)

And finally the conjugate is introduced as

q = p − q = q0 − q1i − q2 j − q3k.

The following statements follow from the definition:

qr = r q, qq = qq = ‖q‖2, ‖qr‖ = ‖rq‖ = ‖q‖‖r‖,〈q, r〉 = 1

2(qr + rq) = (qr) = (qr) ,

〈q, rq〉 = ‖q‖2r. (12)

Moreover, q commutes with any quaternion r if and only if q ∈ R.With this definitionH is a noncommutative field or a division ring that contains

the real numbers as a subfield [9]. It is also true that the complex numbers canbe included in it; however, there are several ways to do it. For any I ∈ H withI2 = −1, we can find a commutative fieldC(I ) = a+bI : a, b ∈ R isomorphicto C, having I as the imaginary unit. This provides some useful information. Takeq ∈ H; then

q = q + ‖q‖ q

‖q‖ , (13)

where I = q‖q‖ belong to

S = I ∈ H : I2 = −1 = q1i + q2 j + q3k : q2

1 + q22 + q2

3 = 1,

so

H =⋃

I∈S

C(I ).


In this way, we can think of H as a bundle of complex planes with the real line asa common axis.

Using Newton’s binomial formula and the commutativity of the product whenone element belongs to R, we also find, for each positive integer n,

qn = (q + q)n =n∑

k=0

(n

k

)(q)k(q)n−k,

which gives easily, using (11),

qn =n/2∑

j=0

(n

2 j

)(−1) j‖q‖2 j(q)n−2 j

+⎛

⎝n/2∑

j=1

(n

2 j − 1

)(−1) j−1‖q‖2 j−2(q)n−2 j+1

⎞

⎠q, (14)

when n is even, and

qn =(n−1)/2∑

j=0

(n

2 j

)(−1) j‖q‖2 j(q)n−2 j +

+⎛

⎝(n+1)/2∑

j=1

(n

2 j − 1

)(−1) j−1‖q‖2 j−2(q)n−2 j+1

⎞

⎠q, (15)

when n is odd.

We also notice that if q : [0, T ] → H and r : [0, T ] → H are differentiable,then, qr is differentiable and

(qr)′(t) = q(t)r ′(t) + q′(t)r(t). (16)

Consequently, for any integer n ≥ 1, qn is differentiable and[qn(t)

]′ = q′(t)qn−1(t) + q(t)[qn−1]′

(t)

= q′(t)qn−1(t) + q(t)[q′(t)qn−2(t) + q(t)

[qn−2]′

(t)]

(17)

= . . . =n−1∑

j=0

q j(t)q′(t)qn− j(t).

If q ∈ H, the exponential of q is defined by

exp q :=∞∑

n=0

qn

n!(the series converges absolutely and uniformly on compact subsets). If r ∈ H issuch that qr = rq, the exponential satisfies the addition theorem

(exp q)(exp r) = exp(q + r).


We also have from (13) the formula

exp q = (exp q)

[(cos ‖q‖)e + (sin ‖q‖) q

‖q‖]

.

Thus exp q = 1 if and only if

q = 0 and ‖q‖ = 0 (mod 2π).

If q : [0, T ] → H is differentiable and if q(t)q′(t) = q′(t)q(t) for all t ∈ [0, T ], itfollows easily from (17) that

[exp q(t)

]′ = [exp q(t)

]q′(t). (18)

Finally, if q is continuous, then

[∫ T

0q(t) dt

]=

∫ T

0q(t) dt,

[∫ T

0q(t) dt

]=

∫ T

0q(t) dt.

3. The monomial linear case

In this section, we first consider the homogeneous monomial linear quaternionicordinary differential equation

q′ = a(t)q (19)

and the corresponding nonhomogeneous one

q′ = a(t)q + f(t), (20)

where a : [0, T ] → H and f : [0, T ] → H are continuous. Let

A : [0, T ] → H, t → A(t) =∫ t

0a(s) ds (21)

denote the indefinite integral of a. We shall first look for conditions upon a(t)implying an explicit expression for the solution of (19) and (20).

Proposition 3.1. Assume that the commutativity property

a(t)A(t) = A(t)a(t), (t ∈ [0, T ]) (22)

holds. Then the solutions of (19) are given by

q(t) = [exp A(t)

]q(0), (t ∈ [0, T ]), (23)

and the solutions of (20) are given by

q(t) = [exp A(t)

]q(0) +

∫ t

0[exp A(t)][exp(−A(s))] f(s) ds

= exp A(t)

(q(0) +

∫ t

0[exp(−A(s))] f(s) ds

), (t ∈ [0, T ]). (24)


Proof. It follows easily from (16) and (18). Remark 3.2. Notice that condition (22) always holds if a(t) = α(t) + β(t)h, whereh ∈ H and α, β : [0, T ] → R are some continuous functions.

Corollary 3.3. If the commutativity condition (22) holds, (20) has a uniqueT-periodic solution for each f (and in particular (19) has only the trivial T-periodicsolution) if and only if a is such that

∫ T

0a(s) ds = 0 or

∥∥∥∥∫ T

0a(s) ds

∥∥∥∥ = 0 (mod 2π). (25)

Proof. Because of (24), q is a T-periodic solution of (20) if and only if q(0) satisfiesthe equation

[exp

(∫ T

0a(s) ds

)− 1

]q(0) =

∫ T

0

[exp A(T )

] [exp(−A(s))

]f(s) ds,

and, asH is a division algebra, this equation is solvable for any right-hand memberif and only if

exp(∫ T

0a(s) ds

)= 1,

i.e., if and only if (25) holds. In a classical way, we can deduce from Corollary 3.3 an explicit formula for

the T-periodic solution of (20) in terms of a Green function.

Corollary 3.4. If conditions (22) and (25) hold, then the unique T-periodic solutionof (20) is given by the formula

q(t) =∫ T

0g(t, s) f(s) ds, (26)

where the Green function

g : [0, T ] × [0, T ] → H, (t, s) → g(t, s)

is defined by

[exp A(t)] [1 − exp A(T )

]−1 [exp(−A(s)], if 0 ≤ s ≤ t,

[exp A(t)] [1 − exp A(T )

]−1 [exp A(T )][exp(−A(s)], if t ≤ s ≤ T.

4. Perturbed monomial linearity

The results of Section 3 can be applied in a classical way to the study of theT-periodic solutions of the semilinear quaternionic differential equation


q′ = a(t)q + c(t, q), (27)

where c : [0, T ] × H → H is continuous. Let C([0, T ],H) denote the (Banach)space of continuous quaternionic functions q : [0, T ] → H endowed with theuniform norm

‖q‖∞ = maxt∈[0,T ]

‖q(t)‖.

If conditions (22) and (25) hold, it follows easily from Corollary 3.4 that theT-periodic solutions of (27) are the solutions of the nonlinear integral equation

q(t) =∫ T

0g(t, s)c(s, q(s)) ds

and hence the fixed points of the nonlinear operator

F : C([0, T ],H) → C([0, T ],H), q →∫ T

0g(·, s)c(s, q(s)) ds.

It is standard to show that this operator is compact on each bounded set ofC([0, T ],H). Hence Schauder’s fixed point theorem easily implies the followingexistence theorem.

Theorem 4.1. Assume that a(t) commutes with A(t) for all t ∈ [0, T ] and verifiesassumption (25), and that c satisfies the growth condition

‖c(t, q)‖ ≤ σ‖q‖ + τ (28)

for all t ∈ [0, T ] and q ∈ H, where σ and τ are nonnegative constants such that

σ maxt∈[0,T ]

∫ T

0‖g(t, s)‖ ds < 1. (29)

Then (27) has at least one T-periodic solution.

We show now how to drop the commutativity condition (22) in Theorem 4.1by reinforcing condition (25).

Theorem 4.2. Assume that a verifies the assumption∫ T

0a(s) ds = 0 (30)

and that c satisfies the growth condition

|[c(t, q)q]| ≤ σ‖q‖2 + τ‖q‖ (31)


σ <

∣∣∣∣1

T

∫ T

0a(s) ds

∣∣∣∣ . (32)



Proof. We use Proposition VI.7 and the following Remark 1 in [23], with

V(q) = 1

2log

(‖q‖2 + 1).

Notice that, for each V ∈ C1(H,R), q, and h in H, we have

〈∇V(q), h〉 = 1

‖q‖2 + 1〈q, h〉 = 1

‖q‖2 + 1(hq). (33)

Hence, using (12) we get

〈∇V(q), a(t)q + c(t, q)〉 = ‖q‖2

‖q‖2 + 1a(t) + [c(t, q)q]

‖q‖2 + 1. (34)

Consequently, using (31) we see that the inequalities

−|a(t)| − σ − τ

2≤ 〈∇V(q), a(t)q + c(t, q)〉 ≤ |a(t)| + σ + τ

2hold for all t ∈ [0, T ] and q ∈ H. Furthermore,

lim sup‖q‖→∞

〈∇V(q), a(t)q + c(t, q)〉

≤ a(t) + lim sup‖q‖→∞

σ‖q‖2 + τ‖q‖‖q‖2 + 1

= a(t) + σ (35)

and

lim inf‖q‖→∞〈∇V(q), a(t)q + c(t, q)〉

≥ a(t) − lim sup‖q‖→∞

σ‖q‖2 + τ‖q‖‖q‖2 + 1

= a(t) − σ. (36)

Hence, if∫ T

0a(s) ds < 0,

we deduce from (35) and (32) that∫ T

0

[lim sup‖q‖→∞

〈∇V(q), a(t)q + c(t, q)〉]

dt ≤∫ T

0[a(s) + σ] ds < 0,

and if∫ T

0a(s) ds > 0,

we deduce from (36) and (32) that∫ T

0

[lim inf‖q‖→∞

〈∇V(q), a(t)q + c(t, q)〉]

dt ≥∫ T

0[a(s) − σ] ds > 0.

Thus, all assumptions of Proposition VI.7 and its Remark 1 in [23] are satisfied. An immediate consequence of Theorem 4.2 is the following variant of Corol-

lary 3.3, where a(t) et A(t) are not assumed to commute.


Corollary 4.3. If a satisfies condition (30), equation (20) has a unique T-periodicsolution for each f (and in particular (19) has only the trivial T-periodic solution).

Remark 4.4. Growth condition (28) trivially implies growth condition (31).

5. Perturbed Ginzburg–Landau nonlinearity

In this section, we obtain a partial extension of Theorem 4.2 to a class of equationsof the form

q′ = a(t)‖q‖rq + c(t, q), (37)

where r > 0, a : [0, T ] → H, and c : [0, T ] × H → H are continuous. Noticethat nonlinearities of the type ‖q‖r q are quaternionic versions of a type of nonlin-earities arising in the Ginzburg–Landau equation, which comes from the theory ofsuperconductivity and has been applied to other problems of physics (see, e.g., [2],p. xvii, [8], p. 286).

Theorem 5.1. Assume that a verifies the assumption

a(t) = 0 for all t ∈ [0, T ] (38)


|[c(t, q)q]| ≤ σ‖q‖r+2 + τ‖q‖ (39)

for all t ∈ R and q ∈ H, where σ and τ are nonnegative constants such that

σ < mint∈[0,T ]

|a(t)|. (40)


Proof. We apply Krasnosel’skii’s method of guiding functions, and in particularProposition VI.6 and its Remark 2 in [23]. Define V : H→ R by

V(q) = ‖q‖2

2.

Using (33), we get, using (12),

〈∇V(q), a(t)‖q‖rq + c(t, q)〉 = ‖q‖r+2a(t) + [c(t, q)q]. (41)

Thus, using (39) and (41) we have, if a(t) > 0 for all t ∈ [0, T ],〈∇V(q), a(t)‖q‖rq + c(t, q)〉 ≥ (a(t) − σ) ‖q‖r+2 − τ‖q‖

for all t ∈ [0, T ] and q ∈ H. Hence, letting a0 = mint∈[0,T ] |a(t)|,〈∇V(q), a(t)‖q‖rq + c(t, q)〉 ≥ (a0 − σ) ‖q‖r+2 − τ‖q‖ > 0

if ‖q‖ ≥ R for some sufficiently large R.


Similarly, if a(t) < 0 for all t ∈ [0, T ], we get

〈∇V(q), a(t)‖q‖rq + c(t, q)〉 ≤ (a(t) + σ) ‖q‖r+2 + τ‖q‖for all t ∈ [0, T ] and q ∈ H. Hence

〈∇V(q), a(t)‖q‖rq + c(t, q)〉 ≤ (−a0 + σ) ‖q‖r+2 + τ‖q‖ < 0

if ‖q‖ ≥ R for some sufficiently large R. Thus all conditions of Proposition VI.6and its Remark 2 in [23] hold. Remark 5.2. It is an open problem to know if Theorem 5.1 still holds if one replacesassumption (38) by the weaker assumption (30).

Now, Krasnosel’skii’s theorem on guiding functions we have used to proveTheorem 5.1 remains valid in the setting of solutions bounded over R (see, e.g.,[18], Theorem 8.6) if we replace everywhere [0, T ] by R. Consequently, the proofof Theorem 5.1 also provides the following result. Consider the equation

q′ = a(t)‖q‖rq + c(t, q), (42)

where a : R→ H and c : R× H are continuous.


inft∈R |a(t)| > 0


|[c(t, q)q]| ≤ σ‖q‖r+2 + τ‖q‖for all t ∈ R and q ∈ H, where σ and τ are nonnegative constants such that

σ < inft∈R |a(t)|.

Then (42) has at least one solution that is bounded over R.

6. Perturbed conjugate monomial nonlinearity

In this section, we consider some perturbations of the monomial equation

q′ = a(t)‖q‖rq p,

where p ≥ 0 is an integer, r ≥ 0 is real, p + r ≥ 1, and a : [0, T ] → H iscontinuous. More explicitly, for c : [0, T ] × H→ H continuous, we consider theproblem of the existence of T-periodic solutions for the equation

q′ = a(t)‖q‖rq p + c(t, q). (43)



a(t) ∈ R \ 0 for all t ∈ [0, T ] (44)


‖c(t, q)‖ ≤ σ‖q‖p+r + τ (45)


σ < mint∈[0,T ]

|a(t)|. (46)


Proof. We use Theorem IV.13 of [23] and introduce the homotopy

q′ = λ[a(t)‖q‖rq p + c(t, q)], λ ∈ ]0, 1]. (47)

If q(t) is a possible solution of (47) for some λ ∈ ]0, 1], then, for any integer0 ≤ m ≤ p, we have

qm(t)q′(t)q p−m(t) = λ[a(t)‖q(t)‖rqm(t)q p(t)q p−m(t) + qm(t)c(t, q(t))q p−m(t)

]

= λ[a(t)‖q(t)‖2p+r + qm(t)c(t, q(t))q p−m(t)

],

and hence

[q p+1]′

(t) =p∑

m=0

qm(t)q′(t)q p−m(t)

= λ

[(p + 1)a(t)‖q(t)‖2p+r +

p∑

m=0

qm(t)c(t, q(t))q p−m(t)

].

Integrating both members over [0, T ] and using (45) and the periodicity of q, weget, with a0 = mint∈[0,T ] |a(t)|,

a0

∫ T

0‖q(t)‖2p+r dt ≤ σ

∫ T

0‖q(t)‖2p+r dt + τ

∫ T

0‖q(t)‖p dt.

Then, using the following simple consequence of Hölder inequality

(1

T

∫ T

0‖q(t)‖µ dt

) 1µ

≤(

1

T

∫ T

0‖q(t)‖ν dt

) 1ν

,

for ν ≥ µ ≥ 1, and letting

‖q‖µ =(

1

T

∫ T

0‖q(t)‖µ dt

) 1µ

,

we get

(a0 − σ)‖q‖2p+r2p+r ≤ τ‖q‖p

2p+r ,


so that

‖q‖2p+r ≤ R1,

where R1 =(

τa0−σ

) 1p+r

. Using this estimate and assumption (45), we easily show,

by a Hölder inequality again, that each possible solution of (47) satisfies an a prioribound of the form

1

T

∫ T

0‖q′(t)‖ dt ≤ R2,

and hence there exists R > 0 such that each possible solution of (47) satisfies theinequality ‖q‖∞ < R. Thus the first condition of Theorem IV.13 in [23] is verifiedwith Ω = B(R), the open ball of center 0 and radius R in C([0, T ],H). Now, ifwe define the map F : H→ H by

F(d) =(

1

T

∫ T

0a(t) dt

)‖d‖rd

p + 1

T

∫ T

0c(t, d) dt,

and if, for some λ ∈ [0, 1], d is a possible solution of the equation

F (d, λ) := (1 − λ)

(1

T

∫ T

0a(t) dt

)‖d‖rd

p + λF(d) = 0, (48)

then, by multiplying both members of this equation by d p and using (45), it iseasy to obtain R3 > 0 such that ‖d‖ ≤ R3 for each possible solution of (48).Consequently, for any open ball B(R) of H with radius R > R3, we have, for theBrouwer degree degB,

degB[F, B(R), 0] = degB[F (·, 0), B(R), 0].

From a result of Eilenberg and Niven [10] (see also [11]) it follows easily thatthe last Brouwer degree above has absolute value equal to p, and hence the resultfollows from Theorem IV.13 in [23]. Remark 6.2. It is an open problem to know if Theorem 6.1 holds without thecondition in (44).

Corollary 6.3. For any integer p ≥ 1, a satisfying the conditions of Theorem 6.1and any continuous functions bm : [0, T ] → H, (0 ≤ m ≤ p − 1), the equation

q′ = a(t)q p +p−1∑

m=0

bm(t)qm ,

has at least one T-periodic solution.


7. Perturbed monomial nonlinearity

In this section we will discuss the existence of T-periodic solutions to quaternionicequations of the form

q′ = α(t)‖q‖r q p + f(t), (49)

where p ≥ 0 is an integer, r ≥ 0 is real, p + r ≥ 1, α : [0, T ] → R, andf : [0, T ] → H are continuous. An important special case is the quaternionicRiccati equation (10). By analogy to the known results on the complex Riccatiequation (5), two questions can be raised for (10) :

(1) Do T -periodic solutions exist when the forcing term f is small?(2) Are there some f for which there exists no T -periodic solution?

Let us call f real if f(t) ∈ R ⊂ H for each t ∈ [0, T ], and let I ∈ H be suchthat I2 = −1.

Proposition 7.1. If f is real, every complex plane C(I ) is invariant for (49).

Proof. If q = a+bI ∈ C(I ) as in (13), then we find that α(t)‖q‖r q p+ f(t) ∈ C(I ),and the vector field of the differential equation is tangent. In a more analytical way,using (14) or (15), we find that (49) can be written equivalently

q′ = α(t)‖q‖r A(t, q), q′ = α(t)‖q‖r B(t, q)q,

for some continuous functions A, B : [0, T ] × H → R, and hence the secondequation is equivalent to

q(t) = q(0) exp

[∫ t

0α(s)‖q(s)‖r B(s, q(s)) ds

],

which shows that if I = q(0)/‖q(0)‖, then q(t) ∈ C(I ) for all t ∈ [0, T ]. As a consequence, we can associate the solutions of (49) with f real to the

solutions of the complex-valued equation

z′ = α(t)|z|r z p + f(t). (50)

Corollary 7.2. Assume f real and let q : J ⊂ R→ H be a quaternionic solutionof (49). Then there exist functions a, b : J → R and I ∈ S such that q(t) =a(t) + b(t)I, t ∈ J and z(t) = a(t) + b(t)i is a solution of (50).

Coming now to the first question, we consider the special case of (49) withα(t) ≡ 1 and with f : [0, T ] → H small as a pertubation of the equation

q′ = ‖q‖rq p. (51)

Theorem 7.3. There exists R0 > 0 such that for all continuous f : [0, T ] → H

such that ‖ f ‖∞ ≤ R0, equation

q′ = ‖q‖rq p + f(t) (52)

has at least one T-periodic solution.


Proof. Going to the components, we can see (52) as a system of the form

q′ = g(q) + f(t), (53)

with q = (q0, q1, q2, q3), g : R4 → R4, and f : [0, T ] → R

4. To apply Corollary 7of [7] (or Corollary 4.4 of [24]) to (53), we have to show the existence of an openball B centered at 0 in R4 such that each possible T-periodic solution of equation

q′ = g(q) (54)

satisfying q(t) ∈ B for all t ∈ [0, T ] is such that q(t) ∈ B for all t ∈ [0, T ], andsuch that

degB[g, B, 0] = 0.

It follows from Corollary 7.2 that (54) only has the trivial T-periodic solution, asis the case for the corresponding complex-valued equation

z′ = |z|rz p. (55)

Indeed, letting z = ρ exp(iθ) in (55), we get the equivalent system for (55)

ρ′ = ρr+p cos(p − 1)θ, θ ′ = ρr+p−1 sin(p − 1)θ,

and the existence of the half-line orbits through the origin given by θ = kπp−1 ,

(k = 0, 1, . . . , 2p − 3) prevents the existence of any nontrivial closed orbit for(55). Consequently, the first assumption is satisfied for any open ball B. Now, ifh(q) = q p,

degB[g, B, 0] = degB[(h, B, 0].As it follows now from a result of Eilenberg-Niven [10,11] that

degB[h, B, 0] = p,

the existence of a T-periodic solution for (49) follows. Remark 7.4. In Theorem 7.3, one can replace f(t) by a perturbation c(t, q) suchthat ‖c(t, q)‖ ≤ R0 for all (t, q) ∈ [0, T ] × H.

Remark 7.5. When r = 0 and p = 2, a more elementary but less direct proof ofTheorem 7.3 can be obtained by computing the Brouwer degree degB[I − P, B, 0],where B ⊂ H ∼= R4 is any fixed open ball around zero and P is Poincare’s operatorfor period T . To do that we use again a pertubation argument. Take ε > 0 andconsider the following perturbation of (51):

q′ = q2 + ε. (56)

It follows also from Corollary 7.2 that (56) has for T-periodic solutions only thetwo constant solutions q(t) = ±√

ε, which are in B for sufficiently small ε. In thiscase the corresponding variational equations are

w′ = ±2√

εw (57)


(for this it is crucial that the periodic solution be real) which, in this case, can beeasily integrated. As a consequence, one obtains that

P′ε(±

√ε) = e±√

εT I,

where Pε is the Poincare operator for the T-periodic solutions of (56), so thatdet[I − P′

ε(±√

ε)] = (1 − e±√εT )4 > 0, and then degB[I − Pε, B, 0] = 2.

Finally, by continuity of topological degree degB[I − P, B, 0] = 2. Now, for ‖ f ‖∞sufficiently small, if Pf denotes the Poincare operator associated to the T-periodicsolutions of (10), we have

degB[I − Pf , B, 0] = degB[I − P, B, 0] = 2,

and the result follows.

Finally, Corollary 7.2 shows that the examples given in [6,20] of complex-valued equations

z′ = z2 + f(t)

without periodic solutions immediately provide examples of real forcing terms ffor which the quaternionic Riccati equation (10) has no T-periodic solution. Thissolves the second question for the quaternionic Riccati equation. It remains openfor the general case of (49). Also, from the results in [5] we can describe thepossible structure of the set of periodic solutions of (10). We can have:

(a) No periodic solution,(b) One real-valued periodic solution,(c) Two real-valued periodic solutions,(d) A sphere of periodic quaternionic-valued solutions,(e) All nonreal solutions periodic (real solutions being not continuable).

Acknowledgement. The first author thanks J. Gomez-Torrecillas for helping him in under-standing the algebraic structure of the quaternions division ring.

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Periodic solutions of quaternionic-valued ordinary differential equations